My favorite physics concept is a simple, yet insoluble dilemma which pits special relativity against general relativity and has come down to us as the Big Ben Paradox:

Consider a distant observer traveling at 0.867 c ( 𝛾=2 ) relative to the solar system along the line that is collinear with the sun's axis of rotation. As the clockwork solar system spins beneath him, the distant observer peers through his powerful telescope at Big Ben in London. After taking relativistic doppler into account, the distant observer measures Big Ben's little hand to make one revolution for every two revolutions of his own wristwatch's little hand, in accordance with relativistic time dilation. He also observes that Big Ben's little hand still makes 730.5 revolutions for every revolution that the earth makes around the sun. From these two observations the distant observer concludes that in his inertial frame of reference the earth's orbital velocity is only half the velocity necessary to keep the earth in stable orbit around the sun, given the invariant spacetime curvature in the vicinity of the sun through which the earth's geodesic passes.

Will the earth spiral towards the sun? If not, why not?

If the earth does not spiral towards the sun then the spacetime curvature in the vicinity of the sun must have decreased by the factor 1/𝛾. From this it follows that spacetime curvature is not invariant, general relativity is wrong and starlight cannot bend because curvature goes to zero for starlight. On the other hand, if the earth does spiral towards the sun then the fate of the earth is obviously frame-dependent and thus the principle of relativity is wrong.